package ‘emc’ · the emc and the toemc algorithm the evolutionary monte carlo (emc; liang and...

Package ‘EMC’February 19, 2015

Type Package

Title Evolutionary Monte Carlo (EMC) algorithm

Version 1.3

Date 2011-12-08

Author Gopi Goswami <[email protected]>

Maintainer Gopi Goswami <[email protected]>

Depends R (>= 1.9.0), mvtnorm, MASS, graphics

Description random walk Metropolis, Metropolis Hasting, parallel tempering, evolution-ary Monte Carlo, temperature ladder construction and placement

License GPL (>= 2)

Repository CRAN

Date/Publication 2011-12-11 17:42:15

NeedsCompilation yes

R topics documented:

evolMonteCarlo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2findMaxTemper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6MetropolisHastings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11parallelTempering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13placeTempers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17print . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21randomWalkMetropolis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22utilsForExamples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

Index 26

1

2 evolMonteCarlo

evolMonteCarlo evolutionary Monte Carlo algorithm

Description

Given a multi-modal and multi-dimensional target density function, a (possibly asymmetric) pro-posal distribution and a temperature ladder, this function produces samples from the target usingthe evolutionary Monte Carlo algorithm.

Below sampDim refers to the dimension of the sample space, temperLadderLen refers to the lengthof the temperature ladder, and levelsSaveSampForLen refers to the length of the levelsSaveSampFor.

Usage

evolMonteCarlo(nIters,temperLadder,startingVals,logTarDensFunc,MHPropNewFunc,logMHPropDensFunc = NULL,MHBlocks = NULL,MHBlockNTimes = NULL,moveProbsList = NULL,moveNTimesList = NULL,SCRWMNTimes = NULL,SCRWMPropSD = NULL,levelsSaveSampFor = NULL,nThin = 1,saveFitness = FALSE,verboseLevel = 0,...)

Arguments

nIters integer > 0.

temperLadder double vector with all positive entries, in decreasing order.

startingVals double matrix of dimension temperLadderLen × sampDim or vector of lengthsampDim, in which case the same starting values are used for every temperaturelevel.

logTarDensFunc function of two arguments (draw, ...) that returns the target density evalu-ated in the log scale.

MHPropNewFunc function of four arguments (temperature, block, currentDraw, ...)that returns new Metropolis-Hastings proposals. See details below on the argu-ment block.

evolMonteCarlo 3

logMHPropDensFunc

function of five arguments (temperature, block, currentDraw, proposalDraw, ...)that returns the proposal density evaluated in the log scale. See details below onthe argument block.

MHBlocks list of integer vectors giving dimensions to be blocked together for sampling.It defaults to as.list(1:sampDim), i.e., each dimension is treated as a block onits own. See details below for an example.

MHBlockNTimes integer vector of number of times each block given by MHBlocks should besampled in each iteration. It defaults to rep(1, length(MHBlocks)). Seedetails below for an example.

moveProbsList named list of probabilities adding upto 1.

moveNTimesList named list of integers ≥ 0.

SCRWMNTimes integer > 0.

SCRWMPropSD double > 0.levelsSaveSampFor

integer vector with positive entries.

nThin integer ≥ 1. Every nThin draw is saved.

saveFitness logical.

verboseLevel integer, a value ≥ 2 produces a lot of output.

... optional arguments to be passed to logTarDensFunc, MHPropNewFunc and logMHPropDensFunc.

Details

MHPropNewFunc and logMHPropDensFunc The MHPropNewFunc and the logMHPropDensFunc arecalled multiple times by varying the block argument over 1:length(MHBlocks), so thesefunctions should know how to generate a proposal from the currentDraw or to evaluate theproposal density depending on which block was passed as the argument. See the examplesection for sample code.

MHBlocks and MHBlockNTimes Blocking is an important and useful tool in MCMC that helpsspeed up sampling and hence mixing. Example: Let sampDim = 6. Let we want to sample di-mensions 1, 2, 4 as one block, dimensions 3 and 5 as another and treat dimension 6 as the thirdblock. Suppose we want to sample the three blocks mentioned above 1, 5 and 10 times in eachiteration, respectively. Then we could set MHBlocks = list(c(1, 2, 4), c(3, 5), 6)and MHBlockNTimes = c(1, 5, 10).

The EMC and the TOEMC algorithm The evolutionary Monte Carlo (EMC; Liang and Wong,2001) algorithm is composed of the following moves:

MH Metropolis-Hastings or mutationRC real crossoverSC snooker crossoverRE (random) exchange

The target oriented EMC (TOEMC; Goswami and Liu, 2007) algorithm has the followingadditional moves on top of EMC:

4 evolMonteCarlo

BCE best chromosome exchangeBIRE best importance ratio exchange

BSE best swap exchangeCE cyclic exchange

The current function could be used to run both EMC and TOEMC algorithms by specifyingwhat moves to employ using the following variables.

moveProbsList and moveNTimesList The allowed names for components of moveProbsListand moveNTimesList come from the abbreviated names of the moves above. For exam-ple, the following specifications are valid:

moveProbsList = list(MH = 0.4,RC = 0.3,SC = 0.3)

moveNTimesList = list(MH = 1,RC = floor(temperLadderLen / 2),SC = floor(temperLadderLen / 2),RE = temperLadderLen)

SCRWMNTimes and SCRWMPropSD The conditional sampling step of the snooker crossover (SC)move is done using random walk Metropolis (RWM) with normal proposals; SCRWMNTimesand SCRWMPropSD are the number of RWM draws and the proposal standard deviation forthe RWM step, respectively. Note these variables are only required if the SC move haspositive probability in moveProbsList or a positive number of times in moveNTimesList.

levelsSaveSampFor By default, samples are saved and returned for temperature level temperLadderLen.The levelsSaveSampFor could be used to save samples from other temperature levels as well(e.g., levelsSaveSampFor = 1:temperLadderLen saves samples from all levels).

saveFitness The term fitness refers to the function H(x), where the target density of interest isgiven by:

g(x) ∝ exp[−H(x)/τmin]

H(x) is also known as the energy function. By default, the fitness values are not saved, butone can do so by setting saveFitness = TRUE.

Value

Below nSave refers to ceil(nIters / nThin). This function returns a list with the followingcomponents:

draws array of dimension nSave× sampDim× levelsSaveSampForLen, if saveFitness = FALSE.If saveFitness = TRUE, then the returned array is of dimension nSave ×(sampDim + 1)× levelsSaveSampForLen; i.e., each of the levelsSaveSampForLenmatrices contain the fitness values in their last column.

acceptRatios matrix of the acceptance rates for various moves used.

evolMonteCarlo 5

detailedAcceptRatios

list of matrices with detailed summary of the acceptance rates for variousmoves used.

nIters the nIters argument.

nThin the nThin argument.

nSave as defined above.

temperLadder the temperLadder argument.

startingVals the startingVals argument.

moveProbsList the moveProbsList argument.

moveNTimesList the moveNTimesList argument.levelsSaveSampFor

the levelsSaveSampFor argument.

time the time taken by the run.

Note

The effect of leaving the default value NULL for some of the arguments above are as follows:

logMHPropDensFunc the proposal density MHPropNewFunc is deemed symmetric.MHBlocks as.list(1:sampDim).

MHBlockNTimes rep(1, length(MHBlocks)).moveProbsList list(MH = 0.4, RC = 0.3, SC = 0.3).moveNTimesList list(MH = 1, RC = mm, SC = mm, RE = nn), where

mm <- floor(nn / 2) and nn <- temperLadderLen.SCRWMNTimes 1, if SC is used.SCRWMPropSD needs to be provided by the user, if SC is used.

levelsSaveSampFor temperLadderLen.

Author(s)

Gopi Goswami <[email protected]>

References

Gopi Goswami and Jun S. Liu (2007). On learning strategies for evolutionary Monte Carlo. Statis-tics and Computing 17:1:23-38.

Faming Liang and Wing H.Wong (2001). Real-Parameter Evolutionary Monte Carlo with Applica-tions to Bayesian Mixture Models. Journal of the American Statistical Association 96:653-666.

See Also

parallelTempering

Examples

## Not run:samplerObj <-

6 findMaxTemper

with(VShapedFuncGenerator(-13579),{

allMovesNTimesList <- list(MH = 1, RC = 2, SC = 2, RE = 4,BIRE = 2, BCE = 2, BSE = 2, CE = 2)

evolMonteCarlo(nIters = 2000,temperLadder = c(15, 6, 2, 1),startingVals = c(0, 0),logTarDensFunc = logTarDensFunc,MHPropNewFunc = MHPropNewFunc,moveNTimesList = allMovesNTimesList,SCRWMNTimes = 1,SCRWMPropSD = 3.0,levelsSaveSampFor = seq_len(4),verboseLevel = 1)

})print(samplerObj)print(names(samplerObj))with(samplerObj,{

print(detailedAcceptRatios)print(dim(draws))par(mfcol = c(2, 2))for (ii in seq_along(levelsSaveSampFor)) {

main <- paste('temper:', round(temperLadder[levelsSaveSampFor[ii]], 3))plot(draws[ , , ii],

xlim = c(-5, 20),ylim = c(-8, 8),pch = '.',ask = FALSE,main = as.expression(main),xlab = as.expression(substitute(x[xii], list(xii = 1))),ylab = as.expression(substitute(x[xii], list(xii = 2))))

}})

## End(Not run)

findMaxTemper Find the maximum temperature for parallel MCMC chains

Description

Multiple MCMC chains based algorithms (e.g., parallel tempering, evolutionary Monte Carlo) needa temperature ladder. This function finds the maximum temperature for constructing the ladder.

Below sampDim refers to the dimension of the sample space, temperLadderLen refers to the lengthof the temperature ladder, and levelsSaveSampForLen refers to the length of levelsSaveSampFor.Note, this function calls evolMonteCarlo, so some of the arguments below have the same nameand meaning as the corresponding ones for evolMonteCarlo. See details below for explanation onthe arguments.

findMaxTemper 7

Usage

findMaxTemper(nIters,statsFuncList,startingVals,logTarDensFunc,MHPropNewFunc,logMHPropDensFunc = NULL,temperLadder = NULL,temperLimits = NULL,ladderLen = 10,scheme = 'exponential',schemeParam = 0.5,cutoffDStats = 1.96,cutoffESS = 50,guideMe = TRUE,levelsSaveSampFor = NULL,saveFitness = FALSE,doFullAnal = TRUE,verboseLevel = 0,...)

Arguments

nIters integer > 0.

statsFuncList list of functions of one argument each, which return the value of the statisticevaluated at one MCMC sample or draw.




logMHPropDensFunc



temperLimits double vector with two positive entries.

ladderLen integer > 0.

scheme character.

schemeParam double > 0.

cutoffDStats double > 0.

cutoffESS double > 0.

8 findMaxTemper

guideMe logical.levelsSaveSampFor



doFullAnal logical.



Details

This function is based on the method to find the temperature range introduced in section 4.1 ofGoswami and Liu (2007).

statsFuncList The user specifies this list of functions, each of which is known to be sensitive tothe presence of modes. For example, if both dimension 1 and 3 are sensitive to presence ofmodes, then one could use:

coord1 <- function (xx) { xx[1] }

coord3 <- function (xx) { xx[3] }

statsFuncList <- list(coord1, coord3)

temperLadder This is the temperature ladder needed for the first stage preliminary run. One caneither specify a temperature ladder via temperLadder or specify temperLimits, ladderLen,scheme and schemeParam. For details on the later set of parameters, see below. Note,temperLadder overrides temperLimits, ladderLen, scheme and schemeParam.

temperLimits temperLimits = c(lowerLimit, upperLimit) is a two-tuple of positivenumbers, where the lowerLimit is usually 1 and upperLimit is a number in [100, 1000]. Ifstochastic optimization (via sampling) is the goal, then lowerLimit is taken to be in [0, 1].

ladderLen, scheme and schemeParam These three parameters are required (along with temperLimits)if temperLadder is not provided. We recommend taking ladderLen in [15, 30]. The allowedchoices for scheme and schemeParam are:

scheme schemeParam======== =============

linear NAlog NA

geometric NAmult-power NAadd-power ≥ 0reciprocal NA

exponential ≥ 0tangent ≥ 0

findMaxTemper 9

We recommended using scheme = 'exponential' and schemeParam in [0.3, 0.5].

cutoffDStats This cutoff comes fromNormal1(0, 1), the standard normal distribution (Goswamiand Liu, 2007); the default value 1.96 is a conservative cutoff. Note if you have more than onestatistic in statsFuncList, which is usually the case, using this cutoff may result in differentsuggested maximum temperatures (as can be seen by calling the print function on the resultof findMaxTemper). A conservative recommendation is that you choose the maximum of thesuggested temperatures as the final maximum temperature for use in placeTempers and laterin parallelTempering or evolMonteCarlo.

cutoffESS a cutoff for the effective sample size (ESS) of the underlying Markov chain ergodicestimator and the importance sampling estimators.

guideMe If guideMe = TRUE, then the function suggests different modifications to alter the settingtowards a re-run, in case there are problems with the underlying MCMC run.

doFullAnal If doFullAnal = TRUE, then the search for the maximum temperature is conductedamong all the levels of the temperLadder. In case this switch is turned off, the search formaximum temperature is done in a greedy (and faster) manner, namely, search is stopped assoon as all the statistic(s) in the statsFuncList find some maximum temperature(s). Note,the greedy search may result in much higher maximum temperature (and hence sub-optimal)than needed, so it is not recommended.

levelsSaveSampFor This is passed to evolMonteCarlo for the underlying MCMC run.

Value

This function returns a list with the following components:

temperLadder the temperature ladder used for the underlying MCMC run.

DStats the D-statistic (Goswami and Liu, 2007) values used to find the maximum tem-perature.

cutoffDStats the cutoffDStats argument.

nIters the post burn-in nIters.levelsSaveSampFor


draws array of dimension nIters× sampDim× levelsSaveSampForLen, if saveFitness = FALSE.If saveFitness = TRUE, then the returned array is of dimension nIters ×(sampDim + 1)× levelsSaveSampForLen; i.e., each of the levelsSaveSampForLenmatrices contain the fitness values in their last column.

startingVals the startingVals argument.intermediate statistics

a bunch of intermediate statistics used in the computation of DStats, namely,MCEsts, MCVarEsts, MCESS, ISEsts, ISVarEsts, ISESS, each being computedfor all the statistics provided by statsFuncList argument.


Note


10 findMaxTemper

logMHPropDensFunc the proposal density MHPropNewFunc is deemed symmetric.temperLadder valid temperLimits, ladderLen, scheme and schemeParam

are provided, which are used to construct the temperLadder.temperLimits a valid temperLadder is provided.


Author(s)


References


See Also

placeTempers, parallelTempering, evolMonteCarlo

Examples

## Not run:coord1 <- function (xx) { xx[1] }ss <- function (xx) { sum(xx) }pp <- function (xx) { prod(xx) }statsFuncList <- list(coord1, ss, pp)maxTemperObj <-

with(VShapedFuncGenerator(-13579),findMaxTemper(nIters = 15000,

statsFuncList = statsFuncList,temperLadder = c(20, 15, 10, 5, 1),startingVals = c(0, 0),logTarDensFunc = logTarDensFunc,MHPropNewFunc = MHPropNewFunc,levelsSaveSampFor = seq_len(5),doFullAnal = TRUE,verboseLevel = 1))

print(maxTemperObj)print(names(maxTemperObj))with(maxTemperObj,{

par(mfcol = c(3, 3))for (ii in seq_along(levelsSaveSampFor)) {



MetropolisHastings 11

}})

## End(Not run)

MetropolisHastings The Metropolis-Hastings algorithm

Description

Given a target density function and an asymmetric proposal distribution, this function producessamples from the target using the Metropolis Hastings algorithm.

Below sampDim refers to the dimension of the sample space.

Usage

MetropolisHastings(nIters,startingVal,logTarDensFunc,propNewFunc,logPropDensFunc,MHBlocks = NULL,MHBlockNTimes = NULL,nThin = 1,saveFitness = FALSE,verboseLevel = 0,...)

Arguments

nIters integer > 0.

startingVal double vector of length sampDim.


propNewFunc function of three arguments (block, currentDraw, ...) that returns newMetropolis-Hastings proposals. See details below on the argument block.

logPropDensFunc

function of four arguments (block, currentDraw, proposalDraw, ...)that returns the proposal density evaluated in the log scale. See details below onthe argument block.



12 MetropolisHastings


saveFitness logical indicating whether fitness values should be saved. See details below.


... optional arguments to be passed to logTarDensFunc, propNewFunc and logPropDensFunc.

Details

propNewFunc and logPropDensFunc The propNewFunc and the logPropDensFunc are called mul-tiple times by varying the block argument over 1:length(MHBlocks), so these functionsshould know how to generate a proposal from the currentDraw or to evaluate the proposaldensity depending on which block was passed as the argument. See the example section forsample code.

MHBlocks and MHBlockNTimes Blocking is an important and useful tool in MCMC that helpsspeed up sampling and hence mixing. Example: Let sampDim = 6. Let we want to sample di-mensions 1, 2, 4 as one block, dimensions 3 and 5 as another and treat dimension 6 as the thirdblock. Suppose we want to sample the three blocks mentioned above 1, 5 and 10 times in eachiteration, respectively. Then we could set MHBlocks = list(c(1, 2, 4), c(3, 5), 6)and MHBlockNTimes = c(1, 5, 10)

saveFitness The term fitness refers to the negative of the logTarDensFunc values. By default,the fitness values are not saved, but one can do so by setting saveFitness = TRUE.

Value


draws matrix of dimension nSave× sampDim, if saveFitness = FALSE. If saveFitness = TRUE,then the returned matrix is of dimension nSave × (sampDim + 1), where thefitness values appear in its last column.

acceptRatios matrix of the acceptance rates.detailedAcceptRatios

matrix with detailed summary of the acceptance rates.




startingVal the startingVal argument.


Note


MHBlocks as.list(1:sampDim).MHBlockNTimes rep(1, length(MHBlocks)).

parallelTempering 13

Author(s)


References

Jun S. Liu (2001). Monte Carlo strategies for scientific computing. Springer.

See Also

randomWalkMetropolis, parallelTempering, evolMonteCarlo

Examples


with(CigarShapedFuncGenerator2(-13579),MetropolisHastings(nIters = 5000,

startingVal = c(0, 0),logTarDensFunc = logTarDensFunc,propNewFunc = propNewFunc,logPropDensFunc = logPropDensFunc,verboseLevel = 2))

print(samplerObj)print(names(samplerObj))with(samplerObj,{

print(detailedAcceptRatios)print(dim(draws))plot(draws,

xlim = c(-3, 5),ylim = c(-3, 4),pch = '.',ask = FALSE,main = as.expression(paste('# draws:', nIters)),xlab = as.expression(substitute(x[xii], list(xii = 1))),ylab = as.expression(substitute(x[xii], list(xii = 2))))

})

## End(Not run)

parallelTempering The parallel Tempering algorithm

Description

Given a multi-modal and multi-dimensional target density function, a (possibly asymmetric) pro-posal distribution and a temperature ladder, this function produces samples from the target usingthe parallel tempering algorithm.

Below sampDim refers to the dimension of the sample space, temperLadderLen refers to the lengthof the temperature ladder, and levelsSaveSampForLen refers to the length of the levelsSaveSampFor.

14 parallelTempering

Usage

parallelTempering(nIters,temperLadder,startingVals,logTarDensFunc,MHPropNewFunc,logMHPropDensFunc = NULL,MHBlocks = NULL,MHBlockNTimes = NULL,moveProbsList = NULL,moveNTimesList = NULL,levelsSaveSampFor = NULL,nThin = 1,saveFitness = FALSE,verboseLevel = 0,...)

Arguments

nIters integer > 0.





logMHPropDensFunc




moveProbsList named list of probabilities adding upto 1.

moveNTimesList named list of integers ≥ 0.levelsSaveSampFor




parallelTempering 15



Details

MHPropNewFunc and logMHPropDensFunc The MHPropNewFunc and the logMHPropDensFunc arecalled multiple times by varying the block argument over 1:length(MHBlocks), so thesefunctions should know how to generate a proposal from the currentDraw or to evaluate theproposal density depending on which block was passed as the argument. See the examplesection for sample code.

MHBlocks and MHBlockNTimes Blocking is an important and useful tool in MCMC that helpsspeed up sampling and hence mixing. Example: Let sampDim = 6. Let we want to sample di-mensions 1, 2, 4 as one block, dimensions 3 and 5 as another and treat dimension 6 as the thirdblock. Suppose we want to sample the three blocks mentioned above 1, 5 and 10 times in eachiteration, respectively. Then we could set MHBlocks = list(c(1, 2, 4), c(3, 5), 6)and MHBlockNTimes = c(1, 5, 10).

The parallel tempering algorithm The parallel tempering (PT; Liang and Wong, 2001) algorithmis composed of the following moves:

MH Metropolis-Hastings or mutationRE (random) exchange

The current function could be used to run the PT algorithm by specifying what moves toemploy using the following variables.moveProbsList and moveNTimesList The allowed names for components of moveProbsList

and moveNTimesList come from the abbreviated names of the moves above. For exam-ple, the following specifications are valid:moveProbsList = list(MH = 0.4,

RE = 0.6)

moveNTimesList = list(MH = 1,RE = temperLadderLen)

levelsSaveSampFor By default, samples are saved and returned for temperature level temperLadderLen.The levelsSaveSampFor could be used to save samples from other temperature levels as well(e.g., levelsSaveSampFor = 1:temperLadderLen saves samples from all levels).

saveFitness The term fitness refers to the function H(x), where the target density of interest isgiven by:

g(x) ∝ exp[−H(x)/τmin]

H(x) is also known as the energy function. By default, the fitness values are not saved, butone can do so by setting saveFitness = TRUE.

Value


16 parallelTempering

draws array of dimension nSave× sampDim× levelsSaveSampForLen, if saveFitness = FALSE.If saveFitness = TRUE, then the returned array is of dimension nSave ×(sampDim + 1)× levelsSaveSampForLen; i.e., each of the levelsSaveSampForLenmatrices contain the fitness values in their last column.

acceptRatios matrix of the acceptance rates for various moves used.detailedAcceptRatios

list of matrices with detailed summary of the acceptance rates for variousmoves used.




temperLadder the temperLadder argument.


moveProbsList the moveProbsList argument.

moveNTimesList the moveNTimesList argument.levelsSaveSampFor



Note


logMHPropDensFunc the proposal density MHPropNewFunc is deemed symmetric.MHBlocks as.list(1:sampDim).

MHBlockNTimes rep(1, length(MHBlocks)).moveProbsList list(MH = 0.4, RC = 0.3, SC = 0.3).moveNTimesList list(MH = 1, RC = mm, SC = mm, RE = nn), where

mm <- floor(nn / 2) and nn <- temperLadderLen.levelsSaveSampFor temperLadderLen.

Author(s)


References

Faming Liang and Wing H.Wong (2001). Real-Parameter Evolutionary Monte Carlo with Applica-tions to Bayesian Mixture Models. Journal of the American Statistical Association 96:653-666.

See Also

evolMonteCarlo

Examples

## Not run:

placeTempers 17

samplerObj <-with(VShapedFuncGenerator(-13579),

parallelTempering(nIters = 2000,temperLadder = c(15, 6, 2, 1),startingVals = c(0, 0),logTarDensFunc = logTarDensFunc,MHPropNewFunc = MHPropNewFunc,levelsSaveSampFor = seq_len(4),verboseLevel = 1))


print(detailedAcceptRatios)print(dim(draws))par(mfcol = c(2, 2))for (ii in seq_along(levelsSaveSampFor)) {



}})

## End(Not run)

placeTempers Place the intermediate temperatures between the temperature limits

Description

Multiple MCMC chains based algorithms (e.g., parallel tempering, evolutionary Monte Carlo) needa temperature ladder. This function places the intermediate temperatures between the minimum andthe maximum temperature for the ladder.

Below sampDim refers to the dimension of the sample space, temperLadderLen refers to the lengthof the temperature ladder, and levelsSaveSampForLen refers to the length of levelsSaveSampFor.Note, this function calls evolMonteCarlo, so some of the arguments below have the same nameand meaning as the corresponding ones for evolMonteCarlo. See details below for explanation onthe arguments.

Usage

placeTempers(nIters,acceptRatioLimits,

18 placeTempers

ladderLenMax,startingVals,logTarDensFunc,MHPropNewFunc,logMHPropDensFunc = NULL,temperLadder = NULL,temperLimits = NULL,ladderLen = 15,scheme = 'exponential',schemeParam = 1.5,guideMe = TRUE,levelsSaveSampFor = NULL,saveFitness = FALSE,verboseLevel = 0,...)

Arguments

nIters integer > 0.acceptRatioLimits

double vector of two probabilities.

ladderLenMax integer > 0.




logMHPropDensFunc



temperLimits double vector with two positive entries.

ladderLen integer > 0.

scheme character.

schemeParam double > 0.

guideMe logical.levelsSaveSampFor





placeTempers 19

Details

This function is based on the temperature placement method introduced in section 4.2 of Goswamiand Liu (2007).

acceptRatioLimits This is a range for the estimated acceptance ratios for the random exchangemove for the consecutive temperature levels of the final ladder. It is recommended that speci-fied range is between 0.3 and 0.6.

ladderLenMax It is preferred that one specifies acceptRatioLimits for constructing the final tem-perature ladder. However, If one has some computational limitations then one could alsospecify ladderLenMax which will limit the length of the final temperature ladder produced.This also serves as an upper bound on the number of temperature levels while placing theintermediate temperatures using the acceptRatioLimits.

temperLadder This is the temperature ladder needed for the second stage preliminary run. One caneither specify a temperature ladder via temperLadder or specify temperLimits, ladderLen,scheme and schemeParam. For details on the later set of parameters, see below. Note,temperLadder overrides temperLimits, ladderLen, scheme and schemeParam.

temperLimits temperLimits = c(lowerLimit, upperLimit) is a two-tuple of positivenumbers, where the lowerLimit is usually 1 and upperLimit is a number in [100, 1000]. Ifstochastic optimization (via sampling) is the goal, then lowerLimit is taken to be in [0, 1].Often the upperLimit is the maximum temperature as suggested by findMaxTemper.

ladderLen, scheme and schemeParam These three parameters are required (along with temperLimits)if temperLadder is not provided. We recommend taking ladderLen in [15, 30]. The allowedchoices for scheme and schemeParam are:

scheme schemeParam======== =============

linear NAlog NA

geometric NAmult-power NAadd-power ≥ 0reciprocal NA

exponential ≥ 0tangent ≥ 0

We recommended using scheme = 'exponential' and schemeParam in [1.5, 2].

guideMe If guideMe = TRUE, then the function suggests different modifications to alter the settingtowards a re-run, in case there are problems with the underlying MCMC run.

levelsSaveSampFor This is passed to evolMonteCarlo for the underlying MCMC run.

Value

This function returns a list with the following components:

finalLadder the final temperature ladder found by placing the intermediate temperatures tobe used in parallelTempering or evolMonteCarlo.

20 placeTempers

temperLadder the temperature ladder used for the underlying MCMC run.acceptRatiosEst

the estimated acceptance ratios for the random exchange move for the consecu-tive temperature levels of temperLadder.

CVSqWeights this is the square of the coefficient of variation of the weights of the importancesampling estimators used to estimate the acceptance ratios, namely, estAcceptRatios.

temperLimits the sorted temperLimits argument.acceptRatioLimits

the sorted acceptRatioLimits argument.

nIters the post burn-in nIters.levelsSaveSampFor


draws array of dimension nIters× sampDim× levelsSaveSampForLen, if saveFitness = FALSE.If saveFitness = TRUE, then the returned array is of dimension nIters ×(sampDim + 1)× levelsSaveSampForLen; i.e., each of the levelsSaveSampForLenmatrices contain the fitness values in their last column.



Note


logMHPropDensFunc the proposal density MHPropNewFunc is deemed symmetric.temperLadder valid temperLimits, ladderLen, scheme and schemeParam

are provided, which are used to construct the temperLadder.temperLimits a valid temperLadder is provided.


Author(s)


References


See Also

findMaxTemper, parallelTempering, evolMonteCarlo

Examples

## Not run:placeTempersObj <-

with(VShapedFuncGenerator(-13579),

print 21

placeTempers(nIters = 10000,acceptRatioLimits = c(0.5, 0.6),ladderLenMax = 50,startingVals = c(0, 0),logTarDensFunc = logTarDensFunc,MHPropNewFunc = MHPropNewFunc,temperLimits = c(1, 5),ladderLen = 10,levelsSaveSampFor = seq_len(10),verboseLevel = 1))

print(placeTempersObj)print(names(placeTempersObj))with(placeTempersObj,{

par(mfcol = c(3, 3))for (ii in seq_along(levelsSaveSampFor)) {



}})

## End(Not run)

print The printing family of functions

Description

The printing family of functions for this package.

Usage

## S3 method for class 'EMC'print(x, ...)## S3 method for class 'EMCMaxTemper'print(x, ...)## S3 method for class 'EMCPlaceTempers'print(x, ...)

Arguments

x an object inheriting from class EMC (generated by functions randomWalkMetropolis,MetropolisHastings, parallelTempering and evolMonteCarlo), EMCMaxTemper

22 randomWalkMetropolis

(generated by function findMaxTemper) or EMCPlaceTempers (generated byfunction placeTempers).

... optional arguments passed to print.default; see its documentation.

Author(s)


See Also

randomWalkMetropolis, MetropolisHastings, parallelTempering, evolMonteCarlo, findMaxTemper,placeTempers

randomWalkMetropolis The Random Walk Metropolis algorithm

Description

Given a target density function and a symmetric proposal generating function, this function pro-duces samples from the target using the random walk Metropolis algorithm.

Below sampDim refers to the dimension of the sample space.

Usage

randomWalkMetropolis(nIters,startingVal,logTarDensFunc,propNewFunc,MHBlocks = NULL,MHBlockNTimes = NULL,nThin = 1,saveFitness = FALSE,verboseLevel = 0,...)

Arguments

nIters integer > 0.

startingVal double vector of length sampDim.


propNewFunc function of three arguments (block, currentDraw, ...) that returns newMetropolis-Hastings proposals. See details below on the argument block.


randomWalkMetropolis 23



saveFitness logical indicating whether fitness values should be saved. See details below.


... optional arguments to be passed to logTarDensFunc and propNewFunc.

Details

propNewFunc The propNewFunc is called multiple times by varying the block argument over1:length(MHBlocks), so this function should know how to generate a proposal from thecurrentDraw depending on which block was passed as the argument. See the example sectionfor sample code.

MHBlocks and MHBlockNTimes Blocking is an important and useful tool in MCMC that helpsspeed up sampling and hence mixing. Example: Let sampDim = 6. Let we want to sample di-mensions 1, 2, 4 as one block, dimensions 3 and 5 as another and treat dimension 6 as the thirdblock. Suppose we want to sample the three blocks mentioned above 1, 5 and 10 times in eachiteration, respectively. Then we could set MHBlocks = list(c(1, 2, 4), c(3, 5), 6)and MHBlockNTimes = c(1, 5, 10)

saveFitness The term fitness refers to the negative of the logTarDensFunc values. By default,the fitness values are not saved, but one can do so by setting saveFitness = TRUE.

Value


draws matrix of dimension nSave× sampDim, if saveFitness = FALSE. If saveFitness =TRUE, then the returned matrix is of dimension nSave × (sampDim + 1), wherethe fitness values appear in its last column.

acceptRatios matrix of the acceptance rates.detailedAcceptRatios

matrix with detailed summary of the acceptance rates.




startingVal the startingVal argument.


Note


MHBlocks as.list(1:sampDim).MHBlockNTimes rep(1, length(MHBlocks)).

24 randomWalkMetropolis

Author(s)


References

Jun S. Liu (2001). Monte Carlo strategies for scientific computing. Springer.

See Also

MetropolisHastings, parallelTempering, evolMonteCarlo

Examples


with(CigarShapedFuncGenerator1(-13579),randomWalkMetropolis(nIters = 5000,

startingVal = c(0, 0),logTarDensFunc = logTarDensFunc,propNewFunc = propNewFunc,verboseLevel = 1))


print(detailedAcceptRatios)print(dim(draws))plot(draws,

xlim = c(-3, 5),ylim = c(-3, 4),pch = '.',ask = FALSE,main = as.expression(paste('# draws:', nIters)),xlab = as.expression(substitute(x[xii], list(xii = 1))),ylab = as.expression(substitute(x[xii], list(xii = 2))))

})

samplerObj <-with(threeDimNormalFuncGenerator(-13579),{

randomWalkMetropolis(nIters = 5000,startingVal = c(0, 0, 0),logTarDensFunc = logTarDensFunc,propNewFunc = propNewFunc,MHBlocks = list(c(1, 2), 3),verboseLevel = 1)

})print(samplerObj)print(names(samplerObj))with(samplerObj,{

utilsForExamples 25

print(detailedAcceptRatios)print(dim(draws))pairs(draws,

pch = '.',ask = FALSE,main = as.expression(paste('# draws:', nIters)),labels = c(as.expression(substitute(x[xii], list(xii = 1))),

as.expression(substitute(x[xii], list(xii = 2))),as.expression(substitute(x[xii], list(xii = 3)))))

})

## End(Not run)

utilsForExamples The utility function(s) for examples

Description

The utility function(s) that are used in the example sections of the exported functions in this pack-age.

Usage

CigarShapedFuncGenerator1(seed)CigarShapedFuncGenerator2(seed)VShapedFuncGenerator(seed)WShapedFuncGenerator(seed)uniModeFuncGenerator(seed)twentyModeFuncGenerator(seed)threeDimNormalFuncGenerator(seed)

Arguments

seed the seed for random number generation.

Value

A list containing the objects to be used as arguments to the exported functions in the respectiveexample sections of this package.

Author(s)


See Also

randomWalkMetropolis, MetropolisHastings, parallelTempering, evolMonteCarlo, findMaxTemper,placeTempers

Index

∗Topic datagenutilsForExamples, 25

∗Topic methodsevolMonteCarlo, 2findMaxTemper, 6MetropolisHastings, 11parallelTempering, 13placeTempers, 17randomWalkMetropolis, 22

∗Topic printprint, 21

CigarShapedFuncGenerator1(utilsForExamples), 25

CigarShapedFuncGenerator2(utilsForExamples), 25

evolMonteCarlo, 2, 6, 10, 13, 16, 17, 20, 22,24, 25

findMaxTemper, 6, 19, 20, 22, 25

MetropolisHastings, 11, 22, 24, 25

parallelTempering, 5, 10, 13, 13, 20, 22, 24,25

placeTempers, 10, 17, 22, 25print, 21

randomWalkMetropolis, 13, 22, 22, 25

threeDimNormalFuncGenerator(utilsForExamples), 25

twentyModeFuncGenerator(utilsForExamples), 25

uniModeFuncGenerator(utilsForExamples), 25

utilsForExamples, 25

VShapedFuncGenerator(utilsForExamples), 25

WShapedFuncGenerator(utilsForExamples), 25

26

package ‘emc’ · the emc and the toemc algorithm the evolutionary monte carlo (emc; liang and...

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